Liouville’s Theory in Linear and Nonlinear Partial Differential Equations

The first of Liouville’s theorems in this monograph is concerned with the linear eigenvalue problem in mathematical physics. Having a summary in the first section, we begin with the applied inverse problem in the second section, to introduce boundary identification of parabolic equations and its ill-posedness to reach the inverse Sturm–Liouville problem. In the third section, the Gel’fand–Levitan– Marchenko (GLM) equation is used to reduce the problem to a study on Goursat problems for hyperbolic equations. We are thus lead to eigenvalue problems in the fourth section, and the intersection property as a striking tool in the theory for one space dimension in the fifth section. We then come back to the inverse Sturm–Liouville problem in the sixth section; uniqueness, reconstruction, and linearization of the Hochstadt equation. The \(\tau \)-function emerges as a Fredholm determinant in the final section, from symmetry of the problem and duality between the boundary conditions; those of Dirichlet and Robin.

The Liouville integral for surfaces of constant Gaussian curvature induces a quantized blowup mechanism of solutions to the Gel’fand or Boltzmann–Poisson equation, as is clarified in the previous chapter. The standard argument to approach this phenomenon, however, is the blowup analysis based on the self-similarity of the equation, which is commonly provided in the fundamental equations of mathematical physics. This property is the invariance under transformation involving independent and dependent variables, which causes a lack of compactness of the family of (approximate) solutions. Breaking down of compactness becomes clear by self-similar transformation, to reach a profile of the structure of total set of solutions from the microscopic viewpoint. Meanwhile, a crucial role is taken by the classification of entire solutions, which traces back to Liouville’s theory on harmonic functions. We thus encounter the third of Liouville’s theorems. The first section is a summary. We begin with several fundamental properties of harmonic functions in the second section, and turn to recent results on semilinear elliptic equations with power nonlinearity in the third section, in accordance with the Liouville property concerning entire solutions. We then generalize the result on exponential nonlinearity obtained in the previous chapter in the final section.

This chapter describes the fourth of Liouville’s theorems, which concerns volume derivatives on deforming domains. This transformation theory provides a mathematical formulation of several physical phenomena involving material transport. Its applications, however, spread, beyond mathematical physics, into various areas in analysis and engineering. The first section is a summary. In the second section, we confirm a fundamental elliptic \(H^1\) regularity on Lipschitz domains, and then turn to \(H^2\) regularity on convex domains in accordance with the second fundamental form. In the third section, we describe the above mentioned Liouville’s theorem and application to the free boundary problem. In the fourth section we turn to the problem of fluid mechanics, associated with both coordinates of Lagrange and Euler, to reach the blowup of the solution to irrotational Euler flow. In the fifth section, we describe an abstract theory of shape optimization. The final section is devoted to Hadamard’s variational formulae for Green’s function of Laplacians. The first volume derivative is used for modeling of fundamental equations in physics. It is to be a basic tool to approach the principle of self-organization in the following chapter, besides the scaling described in the previous chapter. The second volume derivative, on the other hand, is more associated with geometric objects, the second fundamental form of the boundary.

The last of Liouville’s theorems treated in this monograph arises in accordance with classical mechanics. It is an integrability of the Hamilton system under the transformation of sympletic variables. The geometric structure hidden behind it is a suggestion to understand the principle of relaxation of singularities, caused by the skew-symmetric interaction of species and also the symmetry achieved by the action–reaction law in the duality between particle density and field distributions. The first section is a summary. In the second section we observe this fact of singularity cancellation in classical mechanics to reach the notion of a Poisson manifold. In the third section we confirm that the relaxation of the singularity, in the models of multi-species in mathematical biology, emerges from this Poisson structure, particularly in dissipative Lotka–Volterra systems. Motivated by this observation, we treat the reaction–diffusion systems with super-critical growth rate in the fourth section. The fifth section is devoted to the quantized blowup mechanism realized in the 2D Smoluchowski–Poisson equation at three levels of time scale; stationary, in finite time, and in infinite time. This model is a simplified system of chemotaxis, but is concerned, rather more, with the motion of the mean field of many point vorticities in relaxation time, that is, from quasi-equilibrium to equilibrium. This physical background is the origin of recursive hierarchy observed in this model, realized as a collapse and sub-collapse dynamics controlled by a Hamiltonian. There Liouville’s theory of transformation acts as a basic tool. In the final section we turn to a problem in mathematical oncology, the chemotactic paradox. We thus conclude this chapter with a fusion of Liouville’s theory in several areas, that is, the method of Lagrange coordinates and classification of entire solutions, which results in quantized blowup mechanism, duality between field and particle, recursive hierarchy, nonlinear spectral mechanics, space homogenization of isolated systems, emergence from reaction network, and cancellation of singularities in multi-component systems.